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Issue
A&A
Volume 676, August 2023
Article Number L7
Number of page(s) 6
Section Letters to the Editor
DOI https://doi.org/10.1051/0004-6361/202346816
Published online 08 August 2023

© The Authors 2023

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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1. Introduction

Kink waves and oscillations have been frequently reported in solar coronal loops (see e.g., Nakariakov et al. 2021, for a recent review). The energy carried by these waves is of great importance, since it may be associated with the coronal heating problem (e.g., Van Doorsselaere et al. 2020). In transversely structured loops, it is generally believed that the damping of kink oscillations and waves depends on mechanisms such as resonant absorption (e.g., Goossens et al. 2011) and mode coupling (e.g., Pascoe et al. 2010), respectively. Through these processes, azimuthal local Alfvén waves can be induced and, following a phase mixing process (e.g., Heyvaerts & Priest 1983; Browning & Priest 1984; Guo et al. 2019b), the Kelvin-Helmholtz instability (KHI) can be enhanced. The development of transverse wave-induced KH vortices has been confirmed by recent numerical progress (e.g., Terradas et al. 2008; Antolin et al. 2014; Magyar & Van Doorsselaere 2016; Howson et al. 2017a,b). Thus, wave energy dissipation can occur at these small scales (e.g., Karampelas et al. 2017, 2019a,b; Guo et al. 2019a,b, 2023; Shi et al. 2021).

An essential issue is to calculate the energy content of kink waves in the corona. Goossens et al. (2009, 2012) pointed out that kink waves have an Alfvénic property in a thin tube (TT) limit, which is a good approximation for coronal loops that usually exhibit a large aspect ratio. This means that we need to be careful when estimating the energy flux in kink waves since they are different from bulk Alfvén waves (Van Doorsselaere et al. 2008), which are usually considered when computing wave energy in previous observations (e.g., Tomczyk et al. 2007; McIntosh et al. 2011). A detailed energy description of kink Alfvénic waves has been given by Goossens et al. (2013). The spatial variation in the energy of kink Alfvénic waves leads to a significant overestimate of the energy flux compared with previous consideration of bulk Alfvén waves. Van Doorsselaere et al. (2014) further extended this work by considering a volumetric filling factor in a bundle of magnetic flux tubes. They obtained the following expression for the average energy flux of kink waves:

F k = 1 2 f ( ρ i + ρ e ) ( 2 π P obs ) 2 ξ obs 2 v gr , $$ \begin{aligned} F_{\rm k} = \displaystyle \frac{1}{2}f(\rho _{\rm i}+\rho _{\rm e} )\left(\displaystyle \frac{2\pi }{P_{\rm obs}}\right)^2\xi _{\rm obs} ^2{ v}_{\rm gr}, \end{aligned} $$(1)

where f is the filling factor and ρi (ρe) represents the internal (external) loop density. Also, Pobs (ξobs) is the observational wave period (displacement) and vgr represents the group speed. This formula has been used, for example, in Gao et al. (2022) to calculate the total energy of decayless kink oscillations recently observed by the Atmospheric Imaging Assembly (AIA, Lemen et al. 2012). It has also been used in Petrova et al. (2023), Shrivastav et al. (2023), and Li & Long (2023) to calculate the energy flux of kink waves observed by the Extreme Ultraviolet Imager (EUI, Rochus et al. 2020) on board the Solar Orbiter (SO, Müller 2020).

The analytical expressions of energy flux obtained by Goossens et al. (2013) and Van Doorsselaere et al. (2014) need to be modified when considering kink oscillations associated with turbulent structures induced by the KHI. As demonstrated in, for instance, Guo et al. (2020), resonant absorption and phase mixing are ideal linear processes. The eigenmode analyses conducted in Goossens et al. (2013) and Van Doorsselaere et al. (2014) still remain in this linear regime with some assumptions, such as pressureless MHD and a piecewise constant density distribution in the transverse direction. In the nonlinear regime, however, the development of the transverse wave-induced small scales extends the resonant layer from a loop boundary to almost the whole cross-section of the loop (e.g., Karampelas et al. 2017; Guo et al. 2019b), challenging the reliability of the piecewise constant density profile. Magyar & Van Doorsselaere (2016) also found that the dynamics of transverse oscillating coronal loops deviate from linear damping theory due to the development of the KHI. In addition, the consideration of non-zero plasma β in coronal loops enables the analysis of internal energy, which makes a significant contribution to the total energy. The energy carried by kink waves has been calculated in non-zero plasma β studies by, for instance, Geeraerts (2022), Yuan et al. (2023).

Given the importance of energy flux calculation in future studies, especially in observational studies, it is essential to modify the current formula (1). In this paper, we conduct a numerical simulation to mimic the decayless kink oscillations reported in coronal loops and we compare the numerical Poynting flux with the energy flux deduced from Eq. (1). The current formula is modified to make it more reasonable for computing the total energy flux in the nonlinear regime. Section 2 describes our numerical model, including the equilibrium and numerical setup. In Sect. 3, we present the numerical results and forward modelling results, along with the modification of formula in Eq. (1). Section 4 summarizes our findings, ending with some concluding remarks.

2. Model description

We considered a similar three-dimensional model as in Guo et al. (2019b). The magnetic flux tube is density enhanced and embedded in a uniform background corona. The magnetic field is directed along the z-direction. The density profile is given by:

ρ ( x , y ) = ρ e + ( ρ i − ρ e ) ζ ( x , y ) , $$ \begin{aligned} \rho (x,{ y})=\rho _{\rm e}+\left(\rho _{\rm i}- \rho _{\rm e}\right)\zeta (x,{ y}), \end{aligned} $$(2)

with

ζ ( x , y ) = 1 2 { 1 − tanh [ b ( x 2 + y 2 / R − 1 ) ] } , $$ \begin{aligned} \zeta (x,{ y})=\displaystyle \frac{1}{2} \left\{ 1-\tanh \left[b\left(\sqrt{x^2+{ y}^2}/R-1\right) \right] \right\} , \end{aligned} $$(3)

The internal density is ρi = 2.5 × 10−15 g cm−3 and the density ratio is ρi/ρe = 3. The parameter b = 8 gives the width of the inhomogeneous layer l ≈ 0.4R, with R = 1 Mm being the radius of the tube. The loop length, L, is set to be 150 Mm. The temperature is 1 MK throughout the entire computational domain. To maintain the magnetostatic balance, the magnetic field has a slight variation from internal Bi = 50 G to external Be = 50.07 G.

We employed the PLUTO code (Mignone et al. 2007) to solve the time-dependent MHD equations. A parabolic spatial scheme is used for reconstruction, and the numerical fluxes are computed by a Roe Riemann solver. A Runge-Kutta method was used for time marching. The whole computational domain is [−8,8] Mm × [−8,8] Mm × [0,150] Mm. A uniform grid of 100 points was adopted in the z-direction. In the x- and y-directions, 256 stretched grid cells were adopted, respectively. The highest resolution is 20 km in |x, y|≤2 Mm.

The boundary conditions were specified as follows. All components of the velocity at z = L are set to zero, meaning that this loop end is fixed. The remaining variables there have zero gradients. Regarding the z = 0 plane, the z-component of the velocity is set to be antisymmetric, while vx, vy are described by a continuous, dipole-like driver (e.g., Pascoe et al. 2010; Karampelas et al. 2017). This driver is used to excite decayless kink oscillations in the loop. In the internal loop region (r < R), the time-dependent velocity is

v i = v 0 [ cos ( 2 π t P k ) , 0 , 0 ] , $$ \begin{aligned} \mathbf {v} _{\rm i} ={ v}_0\left[\cos \left(\frac{2\pi t}{P_{\rm k}} \right),0,0\right], \end{aligned} $$(4)

where v0 = 4 km s−1 is the amplitude of the driver. The period Pk = 87 s, corresponding to the eigenperiod of the fundamental kink mode (Edwin & Roberts 1983). In the outside loop region, the driver is spatially dependent. It takes the form:

v e = v 0 R 2 cos ( 2 π t P k ) { ( x − x ′ ) 2 − y 2 [ ( x − x ′ ) 2 + y 2 ] 2 , 2 ( x − x ′ ) y [ ( x − x ′ ) 2 + y 2 ] 2 , 0 } , $$ \begin{aligned} \mathbf {v} _{\rm e} ={ v}_0 R^2 \cos \left(\frac{2\pi t}{P_{\rm k}}\right) \left\{ \frac{\left(x-x^\prime \right)^2-{ y}^2}{\left[\left(x-x^\prime \right)^2+{ y}^2\right]^2}, \frac{2\left(x-x^\prime \right){ y}}{\left[\left(x-x^\prime \right)^2+{ y}^2\right]^2},0 \right\} , \end{aligned} $$(5)

where

x ′ = v 0 P k 2 π sin ( 2 π P k t ) . $$ \begin{aligned} x^\prime =\displaystyle \frac{{ v}_0P_{\rm k}}{2\pi } \sin \left(\displaystyle \frac{2\pi }{P_{\rm k}}t\right). \end{aligned} $$(6)

Here, x′ is the displacement of the driver to follow the motion of the footpoint. Similar setups can be found in, for instance, Karampelas et al. (2017), Guo et al. (2019b), Pelouze et al. (2023).

3. Results

The detailed dynamical evolution of the footpoint-driven loop has been extensively discussed in previous works from the literature (e.g., Karampelas et al. 2017, 2019b; Guo et al. 2019a,b). Figure 1 illustrates a snapshot of the density structure after the KHI is fully developed. The loop apex cut shows that the small-scale structures extend over almost the entire loop cross-section. The increasing velocity around the vortices indicates the onset of resonant azimuthal Alfvén waves in the boundary layer. Compared with previous linear models (e.g., Guo et al. 2020), the current resonance layer is distorted and distributed over a larger region of the loop. Therefore, the wave energy that is enhanced in a resonant layer is now spread over the loop cross-section. A previous work conducted by Goossens et al. (2013) concluded that most of the wave energy is confined to the boundary layer, regardless of whether it is a thick boundary layer or a thin boundary (TB) limit. Here, we extend this conclusion by stating that most of the wave energy is confined from the boundary towards the loop centre due to the extension of the KHI eddies across the magnetic field. To clearly show this cross-field effect, the predominantly z-directed magnetic field is presented in the left panel of Fig. 1.

thumbnail Fig. 1.

Snapshot of the isosurface of density showing the loop structure at t = 980 s. The velocity field is represented by blue arrows. Magnetic field lines are overplotted in the left panel. The loop apex cross-cut is shown on the right.

Now, we take this study a further step to examine the energy flux in the current loop model. Since the energy flux given by Van Doorsselaere et al. (2014) is the spatially averaged total energy flux in a loop system, we can thus calculate the input energy flux in the current numerical model for a direct comparison. Before proceeding, it should be noted that the input energy equals the total energy changes in a numerical model. As previously demonstrated in Guo et al. (2019b), Karampelas et al. (2019a), the simulation maintains an energy balance. Therefore, the input energy flux is equivalent to the total energy flux changes in the loop. We compute the input energy flux by considering the Poynting flux at the driven footpoint. The spatially averaged Poynting flux is given by

S ( t ) = 1 A ∫ A S · d A , $$ \begin{aligned} S(t)=\displaystyle \frac{1}{A}\int _A \boldsymbol{S}\cdot \mathrm{d}{\boldsymbol{A}}, \end{aligned} $$(7)

where A represents the surface area of |x|≤2.83R, |y|≤2.83R. S = E × B/μ0 is the Poynting flux, and dA represents the normal surface vector. Figure 2 shows this input energy flux as solid black curves. We note that this energy flux curve starts from zero, although it may not be clearly visible due to the rapid excitation of the kink oscillation in the loop. We can see that the energy flux increases over time – before about 600 s, and then saturation is achieved after around 750 s, indicating that the KHI is fully developed. This scenario has been discussed in, for instance, Karampelas et al. (2019a) and Guo et al. (2019b).

thumbnail Fig. 2.

Energy flux changes in the numerical model and forward model. (a) Energy flux obtained from the numerical results given by Eq. (7) (black line), analytical results given by Eq. (1) (red line), and the energy flux calculated using Eq. (1) with the synthetic displacement in Fig. 3b (blue line). (b) Scatter plots of the energy flux when t > tc. The horizontal axis represents the analytical (red) and synthetic EUI (blue) energy flux, while the vertical axis shows the numerical energy flux. Dashed lines illustrate the linear fits of the data, and their slopes correspond to the factors α and αeui.

We move on to examine the analytical expression in Eq. (1) for the energy flux. As mentioned in Van Doorsselaere et al. (2014), this formula is valid when the filling factor, f, is less than 0.1. Therefore, we chose a computational region of [ − 2.83R, 2.83R]×[−2.83R, 2.83R]×[0, 150R], which gives a filling factor of f = 0.1. This corresponds to the second configuration of a loop system shown in Fig. 2 in Van Doorsselaere et al. (2014). The period, Pobs, is the period of our driver, while the displacement, ξobs, is the observed transverse displacement of the loop. In the following numerical data analysis, we use the displacement of the centre of mass of the loop. The group speed, vgr, is chosen to be the kink speed under the assumption of a long wavelength approximation for a typical coronal loop. The kink speed, ck, is a density-weighted average value (Edwin & Roberts 1983) given by

c k = ρ i v Ai 2 + ρ e v Ae 2 ρ i + ρ e , $$ \begin{aligned} c_{\rm k}=\sqrt{\displaystyle \frac{\rho _{\rm i}{ v}_{\rm Ai}^2 +\rho _{\rm e}{ v}_{\rm Ae}^2}{\rho _{\rm i}+\rho _{\rm e}}}, \end{aligned} $$(8)

where vAi (vAe) represents the internal (external) Alfvén speed. Then, the analytical energy flux computed from Eq. (1) is illustrated in Fig. 2, shown as red lines.

We then compared the numerical energy flux with the analytical results in Fig. 2. When the kink oscillation is not formed, the displacement of the centre of mass is minimal. Therefore, the two curves show different behavior for the initial period. The two curves match well before the KHI is fully developed, indicating that the analytical expression in Eq. (1) is a good description of the energy flux in the linear regime. The deviation is probably caused by the simple transverse density profile considered in the eigenmode analysis. However, when the system enters the nonlinear regime, the formula fails to accurately describe the total energy flux. In this case, a factor of about 1.77 needs to be added to Eq. (1) in order to accurately describe the total energy flux in the nonlinear regime. Therefore, we modify Eq. (1) to obtain:

F k ( t ) = 1 2 α f ( ρ i + ρ e ) ( 2 π P obs ) 2 ξ obs 2 v gr , t > t c , $$ \begin{aligned} F_{\rm k}(t) = \displaystyle \frac{1}{2} \alpha f(\rho _{\rm i}+\rho _{\rm e})\left(\displaystyle \frac{2\pi }{P_{\rm obs}}\right)^2\xi ^2_{\rm obs} { v}_{\rm gr}, ~t>t_{\rm c}, \end{aligned} $$(9)

where α = 1.77 ± 0.01 and tc ∼ 750 s, which indicates the onset time of the KHI. The factor α is the slope of the linear fit of the scatter plots of the numerical energy flux versus the analytical results at each instant when t > tc, as shown in Fig. 2b. The Pearson correlation coefficients are 0.9931 and 0.9903 for the two different fits, respectively.

The parameter α may vary for different numerical setups. However, it is impossible to exhaust all the related parameters. Here, we conducted a parametric survey by changing the amplitude of the footpoint driver to v0 = 6 km s−1. Then, following the above-described procedure, we find α = 2.19 ± 0.02. The factor α is slightly increased for this larger v0 run, due to the larger energy input which leads to a more turbulent loop cross-section, where more energy is confined to the extended resonant layers.

A natural question may arise regarding how we can modify this formula to suit observational analyses. To obtain an observational displacement, we forward-modeled the numerical results using the FoMo code (Van Doorsselaere et al. 2016). To make a comparison with the newest instruments, such as the SO/EUI, we chose the Fe IX 17.1 nm emission line. A similar consideration can be found in Petrova et al. (2023). The numerical results are translated into observables. Figure 3 shows the normalized intensity at the loop apex with a line of sight (LoS) along the y-direction. With this LoS angle, we can observe the largest displacement of the loop in the x-direction. Figure 3a presents the time-distance map with the original numerical resolution. As discussed in Guo et al. (2019b), small structures appear after about four oscillating periods. Following a similar procedure described in Chen et al. (2021), Gao et al. (2022), we degraded the numerical resolution to that of the EUI/HRI at 17.4 nm, which was chosen to have a pixel size of 200 km and a time cadence of 2 s, according to Petrova et al. (2023). The degraded time distance map is shown in Fig. 3b. Although the small structures can not be resolved, we can clearly see the decayless oscillation. The dashed line in Fig. 3b shows the central position of the loop, which can be taken as the observed transverse displacement by EUI.

thumbnail Fig. 3.

Forward-models for the numerical simulation. Upper panel (a) shows the time–distance map of the normalized intensities with the full numerical resolution at the loop apex with a LoS angle along the y-direction. Lower panel (b) shows a degraded resolution result comparable to SO/EUI. The dashed line represents the oscillation profile of the intensities, obtained by calculating the centre of gravity of the intensity in panel (b).

Figure 2 also shows a comparison among the numerical results, the energy flux derived analytically, and the energy flux estimated using the observed displacement by EUI. We can see that the energy flux calculated from the observed displacement in the forward model is much smaller than the real total energy flux in the numerical simulation. Although the resolution of EUI is one of the highest among current EUV imaging instruments, the displacement it observes is still smaller than the real displacement of the loop shown in the simulation with a higher resolution. Therefore, we modified Eq. (1) for the EUI observations as follows:

F k ( t ) = 1 2 α eui f ( ρ i + ρ e ) ( 2 π P obs ) 2 ξ obs 2 v gr , t > t c , $$ \begin{aligned} F_{\rm k}(t) = \displaystyle \frac{1}{2} \alpha _{\rm eui}f(\rho _{\rm i}+\rho _{\rm e})\left(\displaystyle \frac{2\pi }{P_{\rm obs}}\right)^2\xi _{\rm obs}^2{ v}_{\rm gr},~t>t_{\rm c}, \end{aligned} $$(10)

where αeui = 3.46 ± 0.03. For a footpoint driver with an amplitude of v0 = 6 km s−1, αeui = 4.08 ± 0.196. We stress again that Pobs is the observed period of the kink oscillation, ξobs is the displacement obtained from the time–distance map made by forward modelling, and vgr is approximately equivalent to the phase speed, which can be calculated using the loop length and the observed period.

4. Discussion and concluding remarks

We performed a 3D MHD simulation to study delay-less kink oscillations in a magnetic flux tube. The aim is to compare the energy flux with an analytical expression given by Van Doorsselaere et al. (2014) and to identify modifications to this formula to obtain a more accurate estimation of energy flux in transversely oscillating coronal loops. Forward models were also obtained to mimic SO/EUI observations. The analytical formula for the estimation of energy flux has also been modified for real observations. Our main findings can be summarized as follows.

  1. The energy flux formula derived by Van Doorsselaere et al. (2014) is reasonable before the transverse wave-induced KHI is fully developed, that is, in the linear regime.

  2. The analytical formula underestimates the energy flux as the KHI eddies extend the resonance layer across the magnetic field from the loop boundary towards the loop center. Therefore, the formula needs to be modified when the KHI is fully developed. In numerical simulations, the modified expression for the total energy flux is given by:

    F k = 1 2 α f ( ρ i + ρ e ) ( 2 π P obs ) 2 ξ obs 2 v gr , t > t c , $$ \begin{aligned} F_{\rm k} = \displaystyle \frac{1}{2}\alpha f(\rho _{\rm i}+\rho _{\rm e} )\left(\displaystyle \frac{2\pi }{P_{\rm obs}}\right)^2\xi _{\rm obs} ^2{ v}_{\rm gr},~t>t_{\rm c}, \end{aligned} $$(11)

    where α = 1.77 ± 0.01 (α = 2.19 ± 0.02) for a driver of v0 = 4 km s−1 (v0 = 6 km s−1), tc is the onset time of the KHI, f represents the filling factor of a loop, ρi (ρe) represents the internal (external) loop density, Pobs (ξobs) is the observational wave period (displacement), and vgr represents the group speed.

  3. For the SO/EUI observations, the energy flux of kink waves becomes much smaller when employing the previous energy flux computing formula. Therefore, it should also be modified when using the SO/EUI observational data to calculate the energy flux Fk. The updated formula is given by

    F k = 1 2 α eui f ( ρ i + ρ e ) ( 2 π P obs ) 2 ξ obs 2 v gr , t > t c , $$ \begin{aligned} F_{\rm k} = \displaystyle \frac{1}{2} \alpha _{\rm eui}f(\rho _{\rm i}+\rho _{\rm e} )\left(\displaystyle \frac{2\pi }{P_{\rm obs}}\right)^2\xi _{\rm obs} ^2{ v}_{\rm gr},~t>t_{\rm c}, \end{aligned} $$(12)

    where αeui = 3.46 ± 0.03 (αeui = 4.08 ± 0.196) for a driver of v0 = 4 km s−1 (v0 = 6 km s−1).

The modified expressions for both numerical and forward modelling results reveal that the formula given by Van Doorsselaere et al. (2014) underestimates the total energy flux in a transversely oscillating loop with the appearance of the KHI. For previous EUI observations, Petrova et al. (2023), for instance, the energy flux of kink oscillations should be 6.89 kW m−2 (22.59 kW m−2) for the shorter (longer) frequency oscillations if Eq. (10) is considered. This means that the energy flux of the high-frequency kink oscillations reported before can be comparable to the total energy losses (∼10 kW m−2, Withbroe & Noyes 1977) in the active region corona.

In addition, we also degraded the original forward modelling results to roughly match the resolution of SDO/AIA at 17.1 nm, which has a spatial resolution of 1.2 arcsec and a time cadence of 12 s. A similar procedure can be found in (Guo et al. 2019b). Then we can obtain αaia = 3.65 ± 0.09. Note that the value of αaia is slightly larger than αeui due to the lower resolution of AIA. In previous numerical simulations by Karampelas et al. (2019b), there is no clear correlation between the amplitude of kink oscillation observed by AIA and the input energy from a footpoint driver. This probably indicates that the oscillating amplitudes captured by imaging instruments hide a possibly large input energy flux. In the current work, from a different quantitative perspective for each individual simulation, we stress that the energy flux should be larger than analytical expectations. Given the results of Karampelas et al. (2019b) and the current work, we can conclude that the energy flux of kink oscillations in coronal loops is underestimated by the imaging observations.

Apart from varying the amplitude of the velocity driver, we also conducted a higher resolution run. In this case, we increased the highest resolution to 14.8 km in the domain of |x, y|≤2 Mm. We obtained a slightly increased value of α = 1.94 ± 0.009, compared with the original value of α. It is known that changes in energy flux are sensitive to the energy dissipation within the system (e.g., Klimchuk 2015; Prokopyszyn et al. 2019; Howson 2022). However, before the full development of the energy cascading process, the growth of KHI is still closely related to numerical resistivity (e.g., Guo et al. 2019a). Consequently, a smaller numerical resistivity in the higher-resolution run results in a nearly 2% decrease in energy flux. Meanwhile, the higher resolution leads to a slight decrease in the averaged transverse displacement, as finer scales are captured in the new run. Therefore, it is understandable that the value of α becomes slightly greater.

We note that the resonant driver used in the current simulation may seem artificial. To achieve an efficient energy flux, a driver with an eigenfrequency is a common choice. However, from a more realistic perspective, the use of a broadband driver has been discussed, for instance, by Afanasyev et al. (2019), Pagano et al. (2020), Howson & De Moortel (2023). In this case, the resonant component of the broadband driver can be selected by the loop, thus facilitating efficient energy injection. Nevertheless, regardless of the form of the driver employed, the qualitative results of the current study are not expected to change.

Acknowledgments

The authors acknowledge the funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation program (grant agreement No. 724326). TVD was also supported by the C1 grant TRACEspace of Internal Funds KU Leuven and a Senior Research Project (G088021N) of the FWO Vlaanderen. Y.G. acknowledges the support from the China Scholarship Council (CSC) under file No. 202206010018.

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All Figures

thumbnail Fig. 1.

Snapshot of the isosurface of density showing the loop structure at t = 980 s. The velocity field is represented by blue arrows. Magnetic field lines are overplotted in the left panel. The loop apex cross-cut is shown on the right.
In the text
thumbnail Fig. 2.

Energy flux changes in the numerical model and forward model. (a) Energy flux obtained from the numerical results given by Eq. (7) (black line), analytical results given by Eq. (1) (red line), and the energy flux calculated using Eq. (1) with the synthetic displacement in Fig. 3b (blue line). (b) Scatter plots of the energy flux when t > tc. The horizontal axis represents the analytical (red) and synthetic EUI (blue) energy flux, while the vertical axis shows the numerical energy flux. Dashed lines illustrate the linear fits of the data, and their slopes correspond to the factors α and αeui.
In the text
thumbnail Fig. 3.

Forward-models for the numerical simulation. Upper panel (a) shows the time–distance map of the normalized intensities with the full numerical resolution at the loop apex with a LoS angle along the y-direction. Lower panel (b) shows a degraded resolution result comparable to SO/EUI. The dashed line represents the oscillation profile of the intensities, obtained by calculating the centre of gravity of the intensity in panel (b).
In the text

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